I'm a little late in realizing it, but today is Pigeon-hole Day. Festivities include thinking about awesome applications of the Pigeon-hole Principle. So let's come up with some. As always with these kinds of questions, please only post one answer per post so that it's easy for people to vote on them.

Allow me to start with an example:

Brouwer's fixed point theorem can be proved with the Pigeon-hole Principle via Sperner's lemma. There's a proof in Proofs from The Book (unfortunately, the google books preview is missing page 148)

By the way, if you happen to be in Evans at Berkeley today, come play musical chairs at tea!

How strange. I had no idea about pigeon-hole day, but I came here to post a very similar topic (analogues of the pigeonhole principle). But I guess at any given moment, there are lots of people thinking about the pigeonhole principle, and there are only so many websites where such thoughts can reasonably be posted, so applying the...you know.
–
Darsh RanjanNov 5 '09 at 19:18

1

Excellent question. Has anyone asked for a list of applications of the inclusion-exclusion principle? The only sexy one that I know frequently appears in probability courses: the probability that a random permutation has no fixed points converges fairly fast to $1/e$ as the size of the set being permuted grows.
–
Michael HardyJun 13 '10 at 0:10

Kronecker's theorem asserts that if $\lambda$ is irrational then the orbit of $n\lambda$ for $n=1,2,3,\ldots$ is dense in $S^1$ $\simeq$ $\mathbb{R}/\mathbb{Z}$. A proof uses the Pigeon-hole Principle. It relies on the fact that if you divide $S^1$ into $k$ equal (but very small) segments you must hit one of these segments twice, by the Pigeon-hole Principle.

If I'm not mistaken, the original application of the Pigeonhole Principle - the reason we call it Dirichlet's Pigeonhole Principle - was to Dirichlet's Theorem on Diophantine Approximation, viz., if $\alpha$ is a real irrational then there are infinitely many rationals $p/q$ such that $|\alpha-(p/q)|<1/q^2$. An oldie, but still a goodie.

Yes, this was the first application, and Dirichlet then used it (together with an application of the infinite pigeonhole principle) to prove the existence of solutions of the Pell equation. See Supplement VIII of Dirichlet's Vorlesungen ueber Zahlentheorie.
–
John StillwellAug 19 '10 at 22:55

There are a lot of applications of the pigeonhole principle in Ramsey theory. I found a quote by Terence Tao: "Indeed one can view Ramsey thoery as the set of generalizations and repeated applications of the pigeonhole principle." This is from page 254 from the book Additive Combinatorics by Terence Tao and Van Vu.
–
Kristal CantwellNov 6 '09 at 17:41

It looks like the following very important example is not still mentioned here.

Pigeonhole principle plays a crucial role in K. F. Roth's proof that for any $\kappa>2$ and any algebraic irrational real number $\alpha$ inequality $|\alpha-p/q| < q^{-\kappa}$ has only finitely many solutions for rational fractions $p/q$.

Well, the actual author of this application is Siegel (1929). This lemma is known as Siegel's lemma in transcendental number theory. I am surprised it wasn't mentioned earlier. +1
–
Wadim ZudilinDec 3 '10 at 12:58

This lemma itself easily implies Roth's theorem, and so it does not belong to Siegel. But argument does, you are right (Roth's contribution concerns other ideas of the proof).
–
Fedor PetrovDec 3 '10 at 13:29

I remember hearing as an undergrad the "proof" that there are two human beings on the earth with the same number of hairs on their heads. This is done by a few estimations and then applying the pigeonhole principle.

For most cardinals $\kappa \leq \lambda$, it must happen that the infinite symmetric group $S_\kappa$ satisfies exactly the same first order theory as $S_\lambda$. That is, the groups are elementarily equivalent. This is just because there are only continuum many theories in a countable language, but more cardinals than that.

Thue's Lemma, which plays a key role in one proof Fermat's theorem on primes that can be written as the sum of two squares, is based on the pigeonhole principle. (The wikipedia does not mention this and I could not find a nice web page on Thue's Lemma to cite here, so I can only suggest LeVeque's Fundamentals of Number Theory.)

This, of course, requires some heavier theorems in Cech cohomology: If $X$ is a separated scheme that's covered by $d$ affine opens and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then $H^p(X,\mathcal{F}) = 0$ for all $p \geq d$. As a corollary: if $X$ is a quasi-projective scheme over a Noetherian ring $A$, and $\mathcal{F}$ is a quasi-coherent sheaf, then $H^p(X,\mathcal{F}) = 0$ for all $p > d$ where $d = \dim(X)$ (note that $X$ is covered by $d+1$ affines).